ApPROXIMATE LINEAR ANALYSIS OF CONCRETE FRACTURE BY ...
Ba~ant, Z.P., and Cedolin, L. (1984). "Approximate linear analysis of concrete fracture by Rcurves.· J. of Structural Engineering, ASCE, 110, 1336-1355.
ApPROXIMATE LINEAR ANALYSIS OF CONCRETE FRACTURE BY R-CURVES By Zdenek P. Batant,' F. ASCE and Luigi Cedolin; M. ASCE ABSTRACT: Using linear elastic fracture analysis, the energy consumed per unit length of fracture (fracture energy) varies with the crack length, as described by the resistance curve (R-curve). This concept, originally proposed for metals, is developed here into a practical, applicable form for concrete. The energy release rate is determined by an approximate linear elastic fracture analysis based on a certain equivalent crack length, which differs from the actual crack length, and is solved as part of structural analysis. It is shown that such an analysis, coupled with the R-curve concept, allows achieving satisfactory fits of the presently existing fracture data obtained with three-point and four-point bent specimens. Without the R-curve, the use of an equivalent crack length in linear analysis is not sufficient to achieve a satisfactory agreement with these data. The existing data can be described equally well with various formulas for the R-curve, and the material parameters in the formula can vary over a relatively broad range without impairing the representation of test data. Only the overall slope of the R-curve, the initial value, and the fina1 value ,are important. A parabola seems to be the most convenient shape of R-curve because the failure load may then be solved from a quadratic equation. For the general case, a simple algorithm to calculate the failure load is given. Deviations from test data are analyzed statistically, and an approximate relationship of the length parameter of the R-curve to the maximum aggregate size is found.
INTRODUCTION
Due to the large size of the fracture process zone at the crack front, concrete structures do not follow linear elastic fracture mechanics, except when the cross section is extremely large compared to the aggregate size. Nevertheless, engineers need to be able to use linear elastic fracture mechanics at least in some approximate, equivalent sense because nonlinear fracture analysis is much more complicated. Since concrete does not behave plastically under tensile situations, the exterior of the fracture process zone is essentially elastic. Therefore, the stress field farther away from the fracture process zone should be dose to that corresponding to a linear fracture mechanics solution for a certain equivalent crack length. As it turns out, however, this does not suffice to achieve good agreement with fracture test results. Evaluating these results by linear elastic fracture mechanics, one finds that the fracture energy, i.e., the energy consumed by fracture per unit crack length, is variable. Thus, in addition to considering a certain equivalent crack length instead of the actual crack length, one must also take into account the variation of the fracture energy. The situation for concrete happens to be the same as for ductile lProf. of Civ. Engrg. and Dir., Center for Concrete and Geomaterials, Technological Inst., Northwestern Univ., Evanston, m. 60201. ly'isiting Scholar, Northwestern Univ.; Prof. on leave from Dept. of Struct. Engrg., Politecnico di Milano, Italy. Note.-Discussion open until November 1, 1984. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on August 25, 1983. This paper is part of the Journal of Structural Engineering, Vol. 110, No.6, June, 1984. ©ASCE, lSSN 07339445/84/0006-1336/$01.00. Paper No. 18954. 1336
fracture of metals, for which the fracfure energy variation has been Sttid· ied extensively (6). This variation is described by the plot of fracture energy (or fracture toughness) versus the crack extension, c, from a notch or smooth surface. This plot is called the resistance curve or R~e. As is well known, the R~e for any given material cannot be unique unless the crack length, c, is negligible compared to the dimensions of the cross section, the li~ament, and the distance to the nearest applied load. Otherwise, the shape of the R~e depends on these parameters and on the geometrical shape of the stru~ and the nature of loading. The shape of the R~e can be approximately calculated by various methods (for concrete, see Ref. 28, 3). As it appears, however, the shape of the R-
••
n.65
o
WALSH
•
KAPLAN
o
~RV
•
HUANG
If,.
CARPINTERI
*
MINDESS
b. SHAH
•• 0.023
.l-aa •
'i!l{J'~
b. 0.967
~.
..0.038 ... 0.060
0.2
0
.6
I
~.
0.4
Equation 8
13
.
1.4
0.6
TABLE 2.-Valu" of Fracture Energy G," Equation 7
(a)
0.2
0.4
0.6
x.
0.8
1.0
1.2
,.4
Pm/po
FIG. 7.-Examplea of Regressions of Theoretical versus Measured Values of MaxImum Load for Parabolic R-Curve Formula (Eq. 8) and for Various Parameter Values
can be applied. The basic statistical parameters are: s = [~i(Yi - y)2/(n - 2)]1/2 = standard deviation of the deviations Y j - Y from the regression line; n = number of all data points; w = slY = coefficient of variation; in which Y = ~j Yin = ordinate of the centroid of all data points. These statistical parameters, along with optimum slope band Y-intercept Q, are listed in Figs. 7-8. From extensive statistical computer calculations and data plotting it appears that the presently available fracture data for the maximum load can be roughly equally well approximated by both the exponential and parabolic formulas (Eqs. 7-8), and that a relatively wide range of pa1348
,
.,1
o,~ •.• ,
,-
(, ... '"'")
U WA<SH
~"
•
,~.'"
KAPLAN
o GJ\I."
.
•
••
HUANG
.. C .... "".,
1.
~.
•
•
"
.... DOS! 'HAH
-
•• 0",
..
•• Q.(>:"
FIG. ~p. at Regr , i allt_ _ _ _ ........., V.h... " ' _ . ........ I..o.cI for ~ .... A-CurM Fa..." .. (!q. 7)
... meter c~ yield ...... ,Iy equally good results [Fig. 7(0_0»). This finding is the .. me as thai from the aforementioned analysis of the ""pertmental R-: I.
..mn..,
Co!ocIlI8. . . 1. Approximate linear eJastio fracture analysis allows acltieving .. tisfactory fits of exiating fracture data for ronaete. 2. Two .."""tial features are needed, (a) The analys;. must use on equivalent rnu:k length instead of the .ctual notoh or crack length as observed; ond (b) the fracture energy mu.t be considered a•• function 0/ the orad extension (R-curve). 3. The exisling test data CIln be described equaUy well with the uoe of different formulas for the R-auve and different parameter value. in ,~
BalM formula.. The fracture energy value. depend strongly on the type of fonnula used for data evaluation. Thi, doe, nol .eem to be a problem. however. If the same formula is used for both data evaluation and sIructttral anal)lSlS. 4. 'The p:n;ciae .hape of the R~e i. not important from the practical viewpoint, however, who!.re important an! the overaU '!ope. the initial value, and the final value . 5. Coruequently. oophisliisling experimental tedmique •. 6. A parabola is • convenient ""pre.sion for the R.-rurve becallS" il aI10ws solving the failure ]""d from a quadratic equation, If the energy release rate is considered to ""I}' Unearly with the crack length . 7. For an R ao. In the Dugdale-Barenblatt model of crack tip yielding (19) the equivalent crack length can be calculated, but for concrete we can hardly do that because the stress-strain law for the fracture process zone is not known weIl and also the stress distribution is highly random. Therefore, it seems appropriate to consider a - ao to be an empirical function of the fracture process zone size, df . For that, we could use again the estimate df = CI (Kcrlf!i; however, to try another approach, we will exploit the fact that a fully developed fracture process zone has normally the size of several times the maximum aggregate size, da • The use of da is convenient since, in contrast to Kcr and f:, no experiments need be made to determine it. Thus, we assume that a = ao + fda, in which f = empirical parameter to be determined by fitting test data, and ao = actual crack length, interpreted as the notch length in our subsequent fitting of test data. Since the stress distribution along the fracture process zone should depend also on the change of W', we may assume f to depend on 5, and so we set
a = ao + f(S)d a • •••••••••••••••••••••••••••••••••••••••••••••••• (21) and use, instead of Eq. 18, the failure criterion W' (a) = Gf , in which Gf = evaluated at length a, rather than ao. For the definition of 5 (Eq. 17), we now evaluate aW'laa at ao rather than at a, since otherwise Eq. 21 would become an implicit equation in a. For function 1(5), one may simply choose 1(5) =
CI
+ C2 exp (-C35)
........................................
(22)
The values of Gf obtained by fitting the data of Walsh's six concretes are plotted in Fig. lO(a) against the W'-values calculated from measured failure loads. (Note that the Grvalues are also affected by optimization, because they depend on a.) A perfect fit would, in this plot, produce a straight line of slope 1 passing through the origin. The regression line is plotted in Fig. lO(a) as the solid line, and the 95% confidence limits are plotted as the dashed lines. As is seen, the scatter is small. Thus, the hypothesis in Eqs. 21-22 works well for Walsh's data, much better than the previous hypothesis (Eqs. 17-20). Subsequently, other data available in the literature were fitted separately, giving a higher scatter than Walsh's data but still acceptable. However, the values of CI, C2, C3 obtained by optimizations made individuaIly for each data set differ greatly from each other. Moreover, for the combined optimization of all data, the results of which are shown in Fig. lO(b), the exponential decay of the optimized function 1(5) = CI + C2 exp (-C3S) comes out to be so slow (small C3) that 1(5) is almost a straight line. Therefore, the combined optimization of all data sets has also been tried using a simpler linear expression 1(5) = CI + C2S, and the regression analysis yielded about the same standard error. It has also 1353
(b)
/
lib/in}
... .
/
//
0.30
/.
.. /.
//
lib/in)
g
9B
/
gOP
~{. . . . . . r/'//
WALSH
0.20
o.~
/
0
I)
// /
r1 / /
/~...
0.10
0.30
II
/
tliU
/
o
/
D
.•
WALSH
• KAPLAN
•
/
Y
// /
/
//
// Q);/.>/
Gf
/ //-
GJI/lRv HUANG CARPtNTER)
• MINDESS
/
.4
0.20
0.10
SHAH
0.30 (lblin)
W'
FIG. 10.-Equlvalent Crack Length after Eq. 21 and Data Fits Attainable without R-Curves
been tried whether CI , C2' C3 are perhaps some simple functions of concrete properties, rather than constants. In particular, analyses were made with the function f(S) = C1X + C2Y + (C3X + C4Y) 5 and several other more complicated functions, in which x = 1 - 1;;(1,500 psi) and y = 1 - E/ (2 X 107 psi). The rationale behind assuming these functions was the fact that, for very high or very high E, the fracture process zone becomes smaller since the stiffness of the matrix approaches that of the aggregate. However, no improvement in the combined regression of all data could have been achieved in this manner. We must, therefore, conclude that, without the R-curve concept, the representation of test results attainable with a linear elastic analysis is much worse, although still better than that attainable when the actual crack length is considered.
t:
ApPENDIX II.-REFERENCES
1. BaZant, Z. P., "Crack Band Model for Fracture of Geomaterials," Proceedings,
2.
3. 4. 5. 6.
4th International Conference on Numerical Methods in Geomechanics, Edmonton, Alberta, Canada, Vol. 3, Z. Eisenstein, ed., June, 1982, pp. 11371152. BaZant, Z. P., "Mechanics of Fracture and Progressive Cracking in Concrete Structures," Report 83-2/428m, Center for Concrete and Geomaterials, .Northwestern University, Evanston, Ill., Feb., 1983; to ap~ear as Chap~er 1 m.~rac ture Mechanics Applied to Concrete Structures, G. C. Sih, ed., Martinus NIJhoff Publ., The Hague. BaZant, Z. P., and Oh, B. H., "Crack Band Theory for Fracture of Concrete," Materials and Structures (RILEM, Paris), Vol. 16, 1983, pp. 155-177. BaZant, Z. P., and Cedolin, L., "Blunt Crack Band Propagation in Finite Element Analysis," Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM2, Apr., 1979, pp. 297-313. BaZant, Z. P., and Cedolin, L., "Fracture Mechanics of Reinforced Concrete," Journal of the Engineering Mechanics Division, ASCE, Vol. 106, No. EM6, Dec., 1980 (discussion and closure Vol. 108, pp. 464-4,71). . Broek, D., Elementary Engineering Fracture Mechamcs, Noordhoff International Publishing, Leyden, Netherlands, 1974. 1354
7. Brown,]. H., "Measuring the Fracture Toughness of Cement Paste and Mortar," Mlzgazine of Concrete Resetlrch, Vol. 24, No. 81, Dec., 1972. 8. Carpinteri, A., "Ex~rimental Detennination of Fracture Toughness Parameters Krc and JIe for Aggregative Materials," presented at the Mar. 23-Apr. 3, 1981, Sth International Conference on Fracture, Cannes, France, "Advances in Fracture Research," D. Fran~ois, ed., Vol. 4, pp. 1491-1498. 9. Entov, V. M., and Yagust, V. I., "Experimental Investigation of Laws Governing Quasi-Static Development of Macrocracks in Concrete," Mechanics of Solids (translation from Russian), Vol. 10, No.4, 1975, pp. 87-95. 10. GjliYl'V, O. E., Ssrenson, S. I., and Arnesen, A., "Notch Sensitivity and Fracture Toughness of Concrete," Cement and Concrete Research, Vol. 7, 1977, pp. 333-344. 11. Huang, C. M. J., "Finite Element and Experimental Studies of Stress Intensity Factors for Concrete Beams," thesis presented to the Kansas State University, at Kansas, in 1981, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 12. Hutchinson, J. W., and Paris, P. C., "Stability Analysis of J-Controlled Crack Growth; Elastic-Plastic Fracture," ASTM-STF668, American Society for Testing and Materials, 19'79, pp. 37-64. 13. Irwin, G. R., Applied Mathematics Research, Vol. 3, No. 65, 1964. 14. Kaplan, M. F., "Crack Propagation and the Fracture of Concrete," American Concrete Institute Journal, Vol. 58, No. 11, Nov., 1961. 15. Knott, J. F., "Fundamentals of Fracture Mechanics," Butterworth; London, England, 1973. 16. Krafft, J. M., Sullivan, A. M., Boyle, R. W., "Effect of Dimensions on Fast Fracture Instability of Notched Sheets," Cranfield Symposium, 1961, Vol. 1, pp.8-28. 17. Mindess, S., Lawrence, F. V., and Kesler, C. E., "The J-Integral as a Fracture Criterion for Fiber Reinforced Concrete," Cement and Concrete Research, Vol. 7, 1977, pp. 731-742. 18. Paris, P. C., et aI., "The Theory of Tearing Instability of the Tearing Mode of Elastic-Plastic Crack Growth," ASTM Special Technical Publication, No. 668 and 677, 19'79, pp. 5-36. 19. Parker, A. P., "The Mechanics of Fracture and Fatigue," E. &t F.N. Spon, Ltd., London, 1981. . 20. Petersson, P. E., "Fracture Energy of Concrete: Method of Detennination," Cement and Concrete Research, Vol. 10, 1980, pp. 78-89. 21. Petersson, P. E., "Fracture Energy of Concrete: Practical Performance and Experimental Results," Cement and Concrete Research, VoL 10, 1980, pp. 91101. 22. Shah, S. P., and McGarry, F. J., "Griffith Fracture Criterion and Concrete," Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM6, Proc. Paper 8597, Dec., 1971, pp. 1663-1676. 23. Sok, c., and Baron, J., "M~canique de la Rupture Appliqu~ au 'Mton Hydraulique," Cement and Concrete Research, Vol. 9, 1979, pp. 641-648. 24. Tada, H., Paris, P. C., and Irwin, G. R., The Stress Analysis of Cracks Handbook, Del Research Corp., Hellertown, Pa., 1973. 25. Visa1vanich, K., and Naaman, A. E., "Fracture Model for Fiber Reinforced Concrete," Journal of American Concrete Institute, Vol. SO, Mar.-Apr., 1983, pp.128-138. 26. Walsh, P. F., "Fracture of Plain Concrete," The Indian Concrete Journal, Vol. 46, No. 11, Nov., 1979, pp. 469, 470, and 476. 27. Wecharatana, M., and Shah, S. P., "Slow Crack Growth in Cement Composites," Journal of Structural Engineering Division, ASCE, Vol. 108, No. ST6, June, 1982, pp. 1400-1413. 28. Wnuk, M. P., Bdant, Z. P., and Law, E., "Stable Growth of Fracture in Brittle Aggregate Materials," Report, Center for Concrete and Geomaterials, Northwestern University, Evanston, m., Oct., 1983; also to appear in The0retical and Applied Fracture Mechanics, G. Sib, ed., Elsevier-North Honand. 1355
ApPROXIMATE LINEAR ANALYSIS OF CONCRETE FRACTURE BY R-CURVES By Zdenek P. Batant,' F. ASCE and Luigi Cedolin; M. ASCE ABSTRACT: Using linear elastic fracture analysis, the energy consumed per unit length of fracture (fracture energy) varies with the crack length, as described by the resistance curve (R-curve). This concept, originally proposed for metals, is developed here into a practical, applicable form for concrete. The energy release rate is determined by an approximate linear elastic fracture analysis based on a certain equivalent crack length, which differs from the actual crack length, and is solved as part of structural analysis. It is shown that such an analysis, coupled with the R-curve concept, allows achieving satisfactory fits of the presently existing fracture data obtained with three-point and four-point bent specimens. Without the R-curve, the use of an equivalent crack length in linear analysis is not sufficient to achieve a satisfactory agreement with these data. The existing data can be described equally well with various formulas for the R-curve, and the material parameters in the formula can vary over a relatively broad range without impairing the representation of test data. Only the overall slope of the R-curve, the initial value, and the fina1 value ,are important. A parabola seems to be the most convenient shape of R-curve because the failure load may then be solved from a quadratic equation. For the general case, a simple algorithm to calculate the failure load is given. Deviations from test data are analyzed statistically, and an approximate relationship of the length parameter of the R-curve to the maximum aggregate size is found.
INTRODUCTION
Due to the large size of the fracture process zone at the crack front, concrete structures do not follow linear elastic fracture mechanics, except when the cross section is extremely large compared to the aggregate size. Nevertheless, engineers need to be able to use linear elastic fracture mechanics at least in some approximate, equivalent sense because nonlinear fracture analysis is much more complicated. Since concrete does not behave plastically under tensile situations, the exterior of the fracture process zone is essentially elastic. Therefore, the stress field farther away from the fracture process zone should be dose to that corresponding to a linear fracture mechanics solution for a certain equivalent crack length. As it turns out, however, this does not suffice to achieve good agreement with fracture test results. Evaluating these results by linear elastic fracture mechanics, one finds that the fracture energy, i.e., the energy consumed by fracture per unit crack length, is variable. Thus, in addition to considering a certain equivalent crack length instead of the actual crack length, one must also take into account the variation of the fracture energy. The situation for concrete happens to be the same as for ductile lProf. of Civ. Engrg. and Dir., Center for Concrete and Geomaterials, Technological Inst., Northwestern Univ., Evanston, m. 60201. ly'isiting Scholar, Northwestern Univ.; Prof. on leave from Dept. of Struct. Engrg., Politecnico di Milano, Italy. Note.-Discussion open until November 1, 1984. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on August 25, 1983. This paper is part of the Journal of Structural Engineering, Vol. 110, No.6, June, 1984. ©ASCE, lSSN 07339445/84/0006-1336/$01.00. Paper No. 18954. 1336
fracture of metals, for which the fracfure energy variation has been Sttid· ied extensively (6). This variation is described by the plot of fracture energy (or fracture toughness) versus the crack extension, c, from a notch or smooth surface. This plot is called the resistance curve or R~e. As is well known, the R~e for any given material cannot be unique unless the crack length, c, is negligible compared to the dimensions of the cross section, the li~ament, and the distance to the nearest applied load. Otherwise, the shape of the R~e depends on these parameters and on the geometrical shape of the stru~ and the nature of loading. The shape of the R~e can be approximately calculated by various methods (for concrete, see Ref. 28, 3). As it appears, however, the shape of the R-
••
n.65
o
WALSH
•
KAPLAN
o
~RV
•
HUANG
If,.
CARPINTERI
*
MINDESS
b. SHAH
•• 0.023
.l-aa •
'i!l{J'~
b. 0.967
~.
..0.038 ... 0.060
0.2
0
.6
I
~.
0.4
Equation 8
13
.
1.4
0.6
TABLE 2.-Valu" of Fracture Energy G," Equation 7
(a)
0.2
0.4
0.6
x.
0.8
1.0
1.2
,.4
Pm/po
FIG. 7.-Examplea of Regressions of Theoretical versus Measured Values of MaxImum Load for Parabolic R-Curve Formula (Eq. 8) and for Various Parameter Values
can be applied. The basic statistical parameters are: s = [~i(Yi - y)2/(n - 2)]1/2 = standard deviation of the deviations Y j - Y from the regression line; n = number of all data points; w = slY = coefficient of variation; in which Y = ~j Yin = ordinate of the centroid of all data points. These statistical parameters, along with optimum slope band Y-intercept Q, are listed in Figs. 7-8. From extensive statistical computer calculations and data plotting it appears that the presently available fracture data for the maximum load can be roughly equally well approximated by both the exponential and parabolic formulas (Eqs. 7-8), and that a relatively wide range of pa1348
,
.,1
o,~ •.• ,
,-
(, ... '"'")
U WA<SH
~"
•
,~.'"
KAPLAN
o GJ\I."
.
•
••
HUANG
.. C .... "".,
1.
~.
•
•
"
.... DOS! 'HAH
-
•• 0",
..
•• Q.(>:"
FIG. ~p. at Regr , i allt_ _ _ _ ........., V.h... " ' _ . ........ I..o.cI for ~ .... A-CurM Fa..." .. (!q. 7)
... meter c~ yield ...... ,Iy equally good results [Fig. 7(0_0»). This finding is the .. me as thai from the aforementioned analysis of the ""pertmental R-: I.
..mn..,
Co!ocIlI8. . . 1. Approximate linear eJastio fracture analysis allows acltieving .. tisfactory fits of exiating fracture data for ronaete. 2. Two .."""tial features are needed, (a) The analys;. must use on equivalent rnu:k length instead of the .ctual notoh or crack length as observed; ond (b) the fracture energy mu.t be considered a•• function 0/ the orad extension (R-curve). 3. The exisling test data CIln be described equaUy well with the uoe of different formulas for the R-auve and different parameter value. in ,~
BalM formula.. The fracture energy value. depend strongly on the type of fonnula used for data evaluation. Thi, doe, nol .eem to be a problem. however. If the same formula is used for both data evaluation and sIructttral anal)lSlS. 4. 'The p:n;ciae .hape of the R~e i. not important from the practical viewpoint, however, who!.re important an! the overaU '!ope. the initial value, and the final value . 5. Coruequently. oophisliisling experimental tedmique •. 6. A parabola is • convenient ""pre.sion for the R.-rurve becallS" il aI10ws solving the failure ]""d from a quadratic equation, If the energy release rate is considered to ""I}' Unearly with the crack length . 7. For an R ao. In the Dugdale-Barenblatt model of crack tip yielding (19) the equivalent crack length can be calculated, but for concrete we can hardly do that because the stress-strain law for the fracture process zone is not known weIl and also the stress distribution is highly random. Therefore, it seems appropriate to consider a - ao to be an empirical function of the fracture process zone size, df . For that, we could use again the estimate df = CI (Kcrlf!i; however, to try another approach, we will exploit the fact that a fully developed fracture process zone has normally the size of several times the maximum aggregate size, da • The use of da is convenient since, in contrast to Kcr and f:, no experiments need be made to determine it. Thus, we assume that a = ao + fda, in which f = empirical parameter to be determined by fitting test data, and ao = actual crack length, interpreted as the notch length in our subsequent fitting of test data. Since the stress distribution along the fracture process zone should depend also on the change of W', we may assume f to depend on 5, and so we set
a = ao + f(S)d a • •••••••••••••••••••••••••••••••••••••••••••••••• (21) and use, instead of Eq. 18, the failure criterion W' (a) = Gf , in which Gf = evaluated at length a, rather than ao. For the definition of 5 (Eq. 17), we now evaluate aW'laa at ao rather than at a, since otherwise Eq. 21 would become an implicit equation in a. For function 1(5), one may simply choose 1(5) =
CI
+ C2 exp (-C35)
........................................
(22)
The values of Gf obtained by fitting the data of Walsh's six concretes are plotted in Fig. lO(a) against the W'-values calculated from measured failure loads. (Note that the Grvalues are also affected by optimization, because they depend on a.) A perfect fit would, in this plot, produce a straight line of slope 1 passing through the origin. The regression line is plotted in Fig. lO(a) as the solid line, and the 95% confidence limits are plotted as the dashed lines. As is seen, the scatter is small. Thus, the hypothesis in Eqs. 21-22 works well for Walsh's data, much better than the previous hypothesis (Eqs. 17-20). Subsequently, other data available in the literature were fitted separately, giving a higher scatter than Walsh's data but still acceptable. However, the values of CI, C2, C3 obtained by optimizations made individuaIly for each data set differ greatly from each other. Moreover, for the combined optimization of all data, the results of which are shown in Fig. lO(b), the exponential decay of the optimized function 1(5) = CI + C2 exp (-C3S) comes out to be so slow (small C3) that 1(5) is almost a straight line. Therefore, the combined optimization of all data sets has also been tried using a simpler linear expression 1(5) = CI + C2S, and the regression analysis yielded about the same standard error. It has also 1353
(b)
/
lib/in}
... .
/
//
0.30
/.
.. /.
//
lib/in)
g
9B
/
gOP
~{. . . . . . r/'//
WALSH
0.20
o.~
/
0
I)
// /
r1 / /
/~...
0.10
0.30
II
/
tliU
/
o
/
D
.•
WALSH
• KAPLAN
•
/
Y
// /
/
//
// Q);/.>/
Gf
/ //-
GJI/lRv HUANG CARPtNTER)
• MINDESS
/
.4
0.20
0.10
SHAH
0.30 (lblin)
W'
FIG. 10.-Equlvalent Crack Length after Eq. 21 and Data Fits Attainable without R-Curves
been tried whether CI , C2' C3 are perhaps some simple functions of concrete properties, rather than constants. In particular, analyses were made with the function f(S) = C1X + C2Y + (C3X + C4Y) 5 and several other more complicated functions, in which x = 1 - 1;;(1,500 psi) and y = 1 - E/ (2 X 107 psi). The rationale behind assuming these functions was the fact that, for very high or very high E, the fracture process zone becomes smaller since the stiffness of the matrix approaches that of the aggregate. However, no improvement in the combined regression of all data could have been achieved in this manner. We must, therefore, conclude that, without the R-curve concept, the representation of test results attainable with a linear elastic analysis is much worse, although still better than that attainable when the actual crack length is considered.
t:
ApPENDIX II.-REFERENCES
1. BaZant, Z. P., "Crack Band Model for Fracture of Geomaterials," Proceedings,
2.
3. 4. 5. 6.
4th International Conference on Numerical Methods in Geomechanics, Edmonton, Alberta, Canada, Vol. 3, Z. Eisenstein, ed., June, 1982, pp. 11371152. BaZant, Z. P., "Mechanics of Fracture and Progressive Cracking in Concrete Structures," Report 83-2/428m, Center for Concrete and Geomaterials, .Northwestern University, Evanston, Ill., Feb., 1983; to ap~ear as Chap~er 1 m.~rac ture Mechanics Applied to Concrete Structures, G. C. Sih, ed., Martinus NIJhoff Publ., The Hague. BaZant, Z. P., and Oh, B. H., "Crack Band Theory for Fracture of Concrete," Materials and Structures (RILEM, Paris), Vol. 16, 1983, pp. 155-177. BaZant, Z. P., and Cedolin, L., "Blunt Crack Band Propagation in Finite Element Analysis," Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM2, Apr., 1979, pp. 297-313. BaZant, Z. P., and Cedolin, L., "Fracture Mechanics of Reinforced Concrete," Journal of the Engineering Mechanics Division, ASCE, Vol. 106, No. EM6, Dec., 1980 (discussion and closure Vol. 108, pp. 464-4,71). . Broek, D., Elementary Engineering Fracture Mechamcs, Noordhoff International Publishing, Leyden, Netherlands, 1974. 1354
7. Brown,]. H., "Measuring the Fracture Toughness of Cement Paste and Mortar," Mlzgazine of Concrete Resetlrch, Vol. 24, No. 81, Dec., 1972. 8. Carpinteri, A., "Ex~rimental Detennination of Fracture Toughness Parameters Krc and JIe for Aggregative Materials," presented at the Mar. 23-Apr. 3, 1981, Sth International Conference on Fracture, Cannes, France, "Advances in Fracture Research," D. Fran~ois, ed., Vol. 4, pp. 1491-1498. 9. Entov, V. M., and Yagust, V. I., "Experimental Investigation of Laws Governing Quasi-Static Development of Macrocracks in Concrete," Mechanics of Solids (translation from Russian), Vol. 10, No.4, 1975, pp. 87-95. 10. GjliYl'V, O. E., Ssrenson, S. I., and Arnesen, A., "Notch Sensitivity and Fracture Toughness of Concrete," Cement and Concrete Research, Vol. 7, 1977, pp. 333-344. 11. Huang, C. M. J., "Finite Element and Experimental Studies of Stress Intensity Factors for Concrete Beams," thesis presented to the Kansas State University, at Kansas, in 1981, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 12. Hutchinson, J. W., and Paris, P. C., "Stability Analysis of J-Controlled Crack Growth; Elastic-Plastic Fracture," ASTM-STF668, American Society for Testing and Materials, 19'79, pp. 37-64. 13. Irwin, G. R., Applied Mathematics Research, Vol. 3, No. 65, 1964. 14. Kaplan, M. F., "Crack Propagation and the Fracture of Concrete," American Concrete Institute Journal, Vol. 58, No. 11, Nov., 1961. 15. Knott, J. F., "Fundamentals of Fracture Mechanics," Butterworth; London, England, 1973. 16. Krafft, J. M., Sullivan, A. M., Boyle, R. W., "Effect of Dimensions on Fast Fracture Instability of Notched Sheets," Cranfield Symposium, 1961, Vol. 1, pp.8-28. 17. Mindess, S., Lawrence, F. V., and Kesler, C. E., "The J-Integral as a Fracture Criterion for Fiber Reinforced Concrete," Cement and Concrete Research, Vol. 7, 1977, pp. 731-742. 18. Paris, P. C., et aI., "The Theory of Tearing Instability of the Tearing Mode of Elastic-Plastic Crack Growth," ASTM Special Technical Publication, No. 668 and 677, 19'79, pp. 5-36. 19. Parker, A. P., "The Mechanics of Fracture and Fatigue," E. &t F.N. Spon, Ltd., London, 1981. . 20. Petersson, P. E., "Fracture Energy of Concrete: Method of Detennination," Cement and Concrete Research, Vol. 10, 1980, pp. 78-89. 21. Petersson, P. E., "Fracture Energy of Concrete: Practical Performance and Experimental Results," Cement and Concrete Research, VoL 10, 1980, pp. 91101. 22. Shah, S. P., and McGarry, F. J., "Griffith Fracture Criterion and Concrete," Journal of the Engineering Mechanics Division, ASCE, Vol. 97, No. EM6, Proc. Paper 8597, Dec., 1971, pp. 1663-1676. 23. Sok, c., and Baron, J., "M~canique de la Rupture Appliqu~ au 'Mton Hydraulique," Cement and Concrete Research, Vol. 9, 1979, pp. 641-648. 24. Tada, H., Paris, P. C., and Irwin, G. R., The Stress Analysis of Cracks Handbook, Del Research Corp., Hellertown, Pa., 1973. 25. Visa1vanich, K., and Naaman, A. E., "Fracture Model for Fiber Reinforced Concrete," Journal of American Concrete Institute, Vol. SO, Mar.-Apr., 1983, pp.128-138. 26. Walsh, P. F., "Fracture of Plain Concrete," The Indian Concrete Journal, Vol. 46, No. 11, Nov., 1979, pp. 469, 470, and 476. 27. Wecharatana, M., and Shah, S. P., "Slow Crack Growth in Cement Composites," Journal of Structural Engineering Division, ASCE, Vol. 108, No. ST6, June, 1982, pp. 1400-1413. 28. Wnuk, M. P., Bdant, Z. P., and Law, E., "Stable Growth of Fracture in Brittle Aggregate Materials," Report, Center for Concrete and Geomaterials, Northwestern University, Evanston, m., Oct., 1983; also to appear in The0retical and Applied Fracture Mechanics, G. Sib, ed., Elsevier-North Honand. 1355
